Question

If the chord of contact of the point \( P(1, 1) \) with respect to the circle \( S = x^2 + y^2 + 4x + 6y - 3 = 0 \) meet the circle \( S = 0 \) at A and B, then the area of \( \triangle PAB \) is:

• A. \( \frac{216}{25} \)
• B. \( \frac{108}{25} \)
• C. \( \frac{27}{25} \)
• D. \( \frac{54}{5} \)

Hint:

For problems involving the chord of contact, use the general equation of the chord and apply geometry to find the area of the triangle.

Answer 1

The Correct Option is B

Step 1: Equation of the circle.
We are given the equation of the circle as: \[ S = x^2 + y^2 + 4x + 6y - 3 = 0 \] Rewriting this equation in standard form: \[ (x + 2)^2 + (y + 3)^2 = 16 \] This is a circle with center \( (-2, -3) \) and radius \( 4 \).
Step 2: Equation of the chord of contact.
The equation of the chord of contact from the point \( P(1, 1) \) is given by: \[ xx_1 + yy_1 = r^2 \] Substituting \( P(1,1) \), and \( r = 4 \): \[ x + y = 16 \]
Step 3: Area of triangle.
The area of triangle \( PAB \) formed by the chord of contact and the line joining the origin is: \[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \] Substituting the values and solving gives the area: \[ \text{Area of } \triangle PAB = \frac{108}{25} \]